Maximum likelihood detection of MPSK bursts with inserted reference symbols

ABSTRACT

A fast algorithm for performing maximum likelihood detection of data symbols transmitted as phases of a carrier signal.

BACKGROUND OF THE INVENTION

This invention relates generally to the transmission and detection ofdigital data using analog signals, and more particularly the inventionrelates to the detection of phase shift keying (PSK) encoded digitaldata.

The phase of a carrier signal can be used to encode digital data fortransmission. The number of bits represented by a carrier phase symboldepends on the number of phases M of the carrier in an MPSK data burst.

A prior art approach to the detection of data symbols consists of usinga phase locked loop to lock to the reference symbols and then detectingthe data symbols using the phase reference out of the loop. A relatedapproach is to use both reference symbols and remodulated data symbolsto obtain a loop phase reference. These approaches are well known.

Another approach is to form a phase reference using a filteringoperation on the reference symbols, often called pilot symbol aideddemodulation. This approach is essentially the same as the phase lockedloop approach in the sense that the phase locked loop also performs afiltering operation.

The present invention is concerned with maximum likelihood detection ofdata symbols in an MPSK data burst.

SUMMARY AND DESCRIPTION OF THE DRAWING

The present invention presents a fast algorithm to perform maximumlikelihood detection of data symbols.

The FIGURE of the drawing illustrates a flow diagram of one embodimentin implementing the invention

First consider a specific problem which however has all the essentialfeatures of the general problem. Consider that N data symbols s₁, s₂, .. . S_(N) are transmitted at times 1, 2, . . . N, and that a referencesymbol S_(N+1) is transmitted at time N+1. All N+1 symbols are MPSKsymbols, that is, for k=1, . . . N, s_(k) =e^(j)φk, where φk is auniformly distributed random phase taking values in {0,2 π/M, . . . 2π(M-1)/M}, and for k=N+1, reference symbol S_(N+1) is the MPSK symbole^(j0) =1. The N+1 symbols are transmitted over an AWGN channel withunknown phase, modeled by the equation:

    r=se.sup.jθ +n.                                      1)

where r, s, and n are N+1 length sequences whose k^(th) components arer_(k), s_(k), and n_(k), respectively, k=1, . . . N+1. Further, n is thenoise sequence of independent noise samples, r is the received sequence,and φ is an unknown channel phase, assumed uniformly distributed on(-π,π!.

We now give the maximum likelihood decision rule to recover the data s₁,. . . s_(N). For the moment, first consider the problem where we want torecover s=s₁, . . . s_(N+1), where s_(N+1) is assumed to be unknown. Weknow that the maximum likelihood rule to recover s is the s whichmaximizes p(r|s). From previous work, we know that this is equivalent tofinding the s which maximizes η(s), where: ##EQU1## In general, thereare M solutions to (2). The M solutions only differ by the fact that anytwo solutions are a phase shift of one another by some multiple of 2 π/Mmodulo 2 π/M. Now return to the original problem which is to recover thedata s₁, . . . s_(N). The maximum likelihood estimate of s₁, . . . s_(N)must be the first N components of the unique one of the M solutions of(2) whose s_(N+1) component is e^(j0) =1.

An algorithm to maximize (2) when all S_(k), k=1, . . . N+1 are unknownand differentially encoded is given in K. Mackenthun Jr., "A fastalgorithm for multiple-symbol differential detection of MPSK", IEEETrans. Commun., vol 42, no. 2/3/4, pp. 1471-1474, February/March/April1994. Therefore to find the maximum likelihood estimate of s₁, . . .s_(N) when s_(N+1) is a reference symbol, we only need to modify thealgorithm for the case when s_(N+1) is known.

The modified algorithm to find the maximum likelihood estimate s₁, . . .s_(N) of s₁, . . . s_(N) is as follows. Let Φ be the phase vector Φ=(φ₁,. . . φ_(N+1)), where all φ_(k) can take arbitrary values, includingφ_(N+1). If |r_(k) |=0, arbitrary choice of s_(k) will maximize (2).Therefore, we may assume with no loss in generality that |r_(k) |>0,k=1, . . . N. For a complex number γ, let arg γ! be the angle of γ.

Let Φ=(φ₁, . . . φ_(N+1)) be the unique Φ for which:

    arg{r.sub.k e.sup.-jφk !ε 0,2 π/M),

for k=1 . . . N+1.

Define z_(k) by:

    z.sub.k =r.sub.k e.sup.-jθk.                         3)

For each k, k=1, . . . N+1, calculate arg z_(k) !. List the values argz_(k) ! in order, from largest to smallest. Define the function k(i) asgiving the subscript k of z_(k) for the i^(th) list position, i=1, . . .N+1. Thus, we have: ##EQU2## For i=1, . . . N+1, let:

    g.sub.i z.sub.k(i).                                        5)

For i satisfying N+1≦2(N+1), define:

    g.sub.i =e.sup.-j2 π/M g.sub.i-(N+1).                   6)

Calculate: ##EQU3## and select the largest.

Suppose the largest magnitude in (7) occurs for q=q'. We now find thephase vector Φ corresponding to q=q'. Using (3), (5), and (6), with i inthe range of q'≦i≦q'+N, we have: ##EQU4## The evaluation of (8) and (9)gives elements φ_(k)(l),l=1, . . . N+1, in order of subscript valuek(l), we form the sequence φ₁, φ₂, . . . φ_(N+1), which is the vector Φ.The maximum likelihood estimate of s₁, . . . s_(N) is now given by s_(k)=e^(-j)φk, k=1, . . . N, where φ_(k) =φ_(k) -φ_(N+1), k=1, . . . N.

As discussed in Mackenthun supra, algorithm complexity is essentiallythe complexity of sorting to obtain (4), which is (N+1)log(N+1)operations.

We now expand the specific problem considered earlier to a more generalproblem. Suppose that N data symbols are transmitted followed by Lreference symbols s_(N+1), . . . s_(N+L), where s_(k) =e^(j0) =1 fork=N+1, . . . N+L, and assume the definition of channel model (1) isexpanded so that r, s, and n are N+L length sequences. Then in place of(2) we have: ##EQU5## However, note that (10) can be rewritten as:##EQU6## where r'_(N+1) =r_(N+1) +r_(N+2) + . . . r_(N+L). But we canapply the previous modified algorithm exactly to (11) and thereby obtaina maximum likelihood estimate of the first N data symbols.

Now suppose the L reference symbols can take values other than e^(j0).Since the reference symbols are known to the receiver, we can remodulatethem to e^(j0) and then obtain a result in the form (11), and apply theprevious algorithm. Finally, suppose the L reference symbols arescattered throughout the data. It is clear that we can still obtain aresult in the form (11) and apply the previous algorithm.

If desired, sorting can be avoided at the expense of an increase incomplexity in the following way. Fix j, jε{1, . . . N+1}. For k=1, . . .N+1, form r_(j) *r_(k), and let g_(j),k be the remodulation of r_(j)*r_(k) such that g_(j),k ε{0,2 π/M}. Now note that the set in (7) is thesame as the set: ##EQU7## Thus, sorting has been eliminated but formingthe above set requires (N+1)² complex multiplications.

The drawing illustrates a flow diagram of the described embodiment inimplementing the algorithm for maximum likelihood detection of the MPSKdata bursts.

While the invention has been described with reference to a specificembodiment, the description is illustrative of the invention and is notto be construed as limiting the invention. Various modifications andapplications may occur to those skilled in the art without departingfrom the true spirit and scope of the invention as defined by theappended claims.

What is claimed is:
 1. A method of maximum likelihood detection of datasymbols in an MPSK data burst comprising the steps of:(a) identifying NMPSK data symbols s₁, s₂, . . . s_(N) at times 1,2, . . . N along with Lreference symbols S_(N+1) . . . S_(N+L) at time N+1, where s_(k)=e^(j)φk for k=1, . . . N, and φk is uniformly distributed random phasetaking value in {0,2 π|M, . . . 2 π(M-1)|M}, and for k=N+1 up to k=N+Lreference symbol S_(k) is an MPSK symbol e^(j0) =1; (b) transmittingsaid N MPSK symbols and L reference symbols over an AWGN channel withunknown phase and modeled as r=se^(j)φ +n, where r, s, and n are N+Llength sequences whose k^(th) components are r_(k), s_(k), and n_(k)k=1, . . . N+L; and (c) finding S₁, . . . S_(N) which maximizes:##EQU8## where r'_(N+1) =r_(N+1) +r_(N+2) . . . r_(N+L), and L=number ofreference symbols.
 2. The method as defined by claim 1, wherein step (c)includes:(c1) defining Φ as the phase vector Φ=(φ₁, . . . φ_(N+1)), and|r_(k) |>0, k=1, . . . N, and for a complex number of γ, let arg γ! bethe angle of γ; (c2) let Φ=(φ₁, . . . φ_(N+1)) be the unique Φ for which

    arg r.sub.k e.sup.-jφk !ε 0,2 π/M),

for k=1, . . . N+1 and

    z.sub.k =r.sub.k e.sup.-jθk ;

(c3) for each k, k=1, . . . N+1, calculate arg(z_(k)), and list valuesin order, from largest to smallest; (c4) define a function k(i) asgiving a subscript k of z_(k) for the i^(th) list position, i=1, . . .N+1 whereby: ##EQU9## (c5) for i=1, . . . N+1, let

    g.sub.i =z.sub.k(i),

and for i satisfying N+1<i≦2(N+1), define:

    g.sub.i =e.sup.-j2 π/M g.sub.i-(N+1) ;

and (c6) calculate: ##EQU10## (c7) select the largest value in step(c6).
 3. The method as defined by claim 2, wherein the largest value instep (c7) occurs for q=q', and further including the steps of:(d)finding a phase vector Φ corresponding to q=q' as follows: ##EQU11## 4.The method as defined by claim 3, wherein step (d) includes arrangingelements φ_(k)(l), l=1, . . . N+1, in order of subscript value k(l), andforming the sequence φ₁, φ₂, . . . φ_(N+1), as the vector φ, the maximumlikelihood sequence s₁, . . . s_(N) being s_(k) =e^(-j)φk, k=1, . . . N,where φ_(k) =φ_(k) -φ_(N+1), k=1, . . . N.
 5. The method as defined byclaim 4, wherein N data symbols are transmitted followed by L referencesymbols s_(N+1), . . . s_(N+L), where s_(k) =e^(j0) =1 for k=N+1, . . .N+L, and r, s, and n are N+L length sequences.
 6. The method as definedby claim 5, wherein j is fixed, jε{1, . . . N+1} and for k=1, . . . N+1,form r_(j) *r_(k), and let g_(j),k be the remodulation of r_(j) *r_(k)such that g_(j),k ε{0,2 π/M} and step c6) becomes: